1. Introduction

In sample surveys, researchers generally estimate the population mean as a measure of location. Many estimators like mean per unit estimator, usual ratio estimator, usual product estimator etc are available in literature to estimate population mean under SRS. These estimators are used because mean is generally more stable measure of location as compared to the others. But, in some situations, median and mode are also more important than the mean. Median is often used as the measure of location for skewed distribution because it is insensitive to extreme values. In this direction, Gross (1980) for the first time proposed sample median as a consistent estimator of population median. Further, Kuk and Mak (1989) suggested ratio estimator for population median. Moreover, a number of estimators for population median have been proposed by the researchers in last two decades.

Mode is another measure of location which has been neglected so far in sample surveys. But in case of unimodal distribution, unique mode represents most likely value of the population. Due to this property, mode sometimes becomes more appropriate measure of location as compared to mean and median. It is especially useful in finding the most popular size in studies related to marketing, trade and industry etc. For example, in the manufacturing of shoes or ready-made garments, one is interested to know which size of shoe or garment is most popular. Also, in the medical field, a medical researcher would normally be more interested in the most probable value of the heart rate than in the total of heart rate measurements divided by the number of measurements.

In literature, very few researchers paid the attention for estimation of population mode, all using different techniques. Initially, Parzan (1962) estimated the mode first estimating the probability density function of the population and then mode. Bickel (2010) estimated the mode first transforming the data to the approximate normal and then estimated the mean which is equals to the mode for the transformed data. Chen et al. (2014) defined the sequence of (k, ρ)-modes where a point is (k, ρ)-mode iff its probability density is higher than all the points within distance k under a distance metric ρ. Recently Sedory and Singh (2014), first estimated the median using technique of Kuk and Mak (1989) then using the empirical relationship for mean, median and mode, defined the estimator of mode.

In this paper, we develop the new simple and efficient technique of estimating the population mode by using well known measures of skewness (Karl Pearson and based upon central moments). Here we first establish the new parametric relationship for population mode involving population parameters and an unknown constant . By proposing the conventional consistent estimator of population mode , we find the approximate value of by minimizing the , upto terms of order . Further, for the bivariate -population, we have also proposed ratio-type and product-type estimators of . We have checked authenticity of the new parametric relationship for . Upto terms of order , the optimum biases and minimum MSE's of the proposed estimators are obtained and compared with each other. Empirically, we have also checked the authenticity of the parametric relationship by showing that for the given population established new parametric relation is more accurate than the empirical relationship for mean, median and mode, and obtained the values of minimum MSE's of the proposed estimators.

2. Notations and Expectations

Suppose a simple random sample of size is drawn from a finite population of size without replacement and observations on both study variables and auxiliary variable are taken. Let the values of variable and be denoted by and respectively on the unit of the population and the corresponding small letters and denote the values corresponding to the unit in the sample.

Taking,

Obviously

Defining,

For the sake of simplicity, assume that is large enough as compares to so that finite population correction (fpc) terms are ignored throughout.

For the given SRSWOR, we have the following expectations,

and up to terms of order

3. Proposed Estimators and Their Biases and MSE’s

Karl Pearson coefficient of skewness and coefficient of skewness based upon moments are respectively given by

(3.1)

and

(3.2)

Without loss of generality, we can take

where is unknown constant to be determined.

Therefore,

That implies

(3.3)

which is a new parametric functional relationship for population mode for any population.

We here propose three estimators of under the three different situations as

(3.4)

(3.5)

(3.6)

where are unknown constants, whose values are determined by minimizing the MSE’s of respective estimators and . Here the estimator uses no information on auxiliary variable which is highly correlated with , whereas and uses the known information of , which are of ratio and product type estimators respectively.

To find the biases and MSE’s of , we expand them in terms of and then taking the expectations as given in section 2, upto terms of order , we get,

and

The above MSE’s of and are minimised for

and the Biases and minimum MSE’s are given as,

and

Where,

Srivastava and Jhajj (1983) have shown that if we replace the parameters involved in the optimum value of the unknown constant by their consistent estimators then upto terms of order , the MSE remains the same. Using this, if we replace the parameters involved in and by their consistent estimators then their estimators say become and respectively. Then our estimators reduce to

and

Among the three estimators, the most efficient is iff

(3.7)

(3.8)

(3.9)

Above conditions (3.7), (3.8) and (3.9) represent the optimum situations for respectively.

4. Authenticity of Parametric Functional Relationship for

The new parametric functional relationship for is

(4.1)

where is an unknown constant whose value is to be determined for the given population.

If is optimised by minimising the MSE of conventional consistent estimator (up to the order ), the above functional relationship reduced to

As in sampling theory, the nature of the population under study is not known. To see the authenticity of approximate parametric relation , we have to consider the population of particular nature. The approximation is due to the fact that we are using the optimum value of unknown constant , not the exact value. When the population is univariate, one can check the relationship for easily. But, if the population is , then it is very difficult check this relationship because for this we need the bivariate moments and marginal distributions of and . No doubt, this relationship can be checked by researchers in inference. So this problem is open for researchers involved in statistical Inference. But in sampling theory, for the known value of we can increase the precision of the estimators and . Following particular populations are considered to check the authenticity of :

(i) Population is Normal:

(a) Let , then we know and . Using these values, we get,

Because in numerator of and also,

which is variance of when fpc are ignored. So is verified for univariate normal population.

(b) Let , then we know is odd. Also, and . Using these values, we get,

The expressions for MSE’s are

and

So, is verified for bivariate normal population and upto terms of order , we have

and

So the relation is true for univariate as well as for bivariate normal populations and in general true for all symmetrical populations.

(ii) Population is Poisson:

Let . We then have so Poisson distribution is always positively skewed, Using these values, we get , so and if is not integer and .

Hence this relation is also verified for positively skewed distributions.

(iii) Population is :

Let , then we have (positively skewed).

(iv) Particular cases:

Hence we conclude that the new functional relationship for is always true for symmetric populations whether univariate or bivariate. In general, it is approximately true for all populations, whereas for the skewed distribution, it is approximately true. The approximation is due to the fact that we use the optimum value of not the exact one.

5. Numerical Illustrations

To illustrate the result numerically, we have made computations for 10 populations taken from literature by using Microsoft Excel 2010.

The source of the populations, the nature of the variables, the values of are listed in Table 1.

In Table 2, we have given the true value of population Mode , the values of the population mode obtained by using the new parametric functional relationship of and values of population mode obtained by using the empirical relationship ‘Mode =3 Median- 2 mean’ for different populations.

The efficiencies of proposed estimators are given in Table 3.

From Table 2, we observe that the values obtained from the new parametric functional relationship for mode i.e. , are close to the true value of the population mode , as compared to the values of population mode obtained by using the empirical relationship for it. The small amount of errors is due to the fact that we are using the optimum value of not the exact value. So, empirically, this relation is verified.

From Table 3, at the end, we observe that in the sampling theory, for the bivariate population, the known value of increases the accuracy of the estimators of population mode . We note that the MSE’s of all are less than , at the same time MSE’s of are less than respectively. When the conditions (3.7) and (3.8) are satisfied then are more efficient than respectively, otherwise is more efficient than the others.

Table 1. Description of populations

Table 2. Values of population mode      , values obtained from new parametric relationship       and empirical relationship

Table 3.

6. Conclusions

The empirical study shows that the new parametric relationship for population mode is more beneficial for estimating population mode than the empirical relationship between mean, median and mode. Also, the values of MSEs of the proposed estimators for mode are less than the values of MSEs of mean per unit estimator of population mean. It means that the proposed estimators of population mode are more stable than the mean per unit estimator of population mean. Further, the conditions have been found out where ratio-type and product-type estimators of population mode are more efficient than . Also, the new parametric relationship for population mode provides a new way to estimate population mode for further study.